Antenna Model of Wind Turbine Struck by Lightning

Damir Cavka, Dragan Poljak Department of Electronics FESB, University of Split

Split, Croatia davka@fesb.hr, dpoljak@fesb.hr

Ranko Goic Department of Power Engineering

FESB, University of Split Split, Croatia rgoic@fesb.hr

AbstractThe paper deals with an analysis of direct lightning strike to the wind turbine using thin wire antenna theory. Wind turbine is represented by a simple wire structure, while the lightning channel is represented by an equivalent lossy vertical wire attached to the wind turbine. The entire structure is energized by an ideal voltage source at the junction point. The analysis is based on the solution of the set of Pocklington integro-differential equations for arbitrary wires above the ground in the frequency domain. The corresponding transient response is obtained by means of Inverse Fourier transform. The set of Pocklington integro-differential equations is solved by the Galerkin-Bubnov Indirect Boundary Element Method.

Keywords-Wind turbine; lightning strike; return stroke; antenna model; GB-IBEM; real ground.

I. INTRODUCTION The wind turbines (WT) are extremely vulnerable to

lightning strikes due to their special shape and isolated locations mainly in high altitude areas. Namely, due to large tower heights and open-air structures placed in often mountainous conditions, wind turbines are very often struck by lightning. Thus, lightning strikes may cause serious damages to wind-turbine components. Available relevant statistics indicates that between 4% and 8% of wind power in Europe suffers damages due to lightning strikes each year [1]. Therefore, there is an objective need for a better insight of lightning discharges impact on WT. One aspect of the problem is the current distribution along the WT in the event of lighting strike.

Generally, in the last few decades an interaction of lightning with tall structures has occupied considerable attention of many researchers (e.g. [2]). Several models have been proposed for the determination of the current distribution along the structure and lightning channel. Usually, models are built from some return stroke models initially developed for the case of return stroke initiated at the ground level. The presence of a tall structure has been included in two types of return stroke models: engineering models and electromagnetic (antenna theory - AT) models [3]. The presence of an elevated object in engineering models has been considered by assuming the object to be a uniform, lossless transmission line [4], [5]. In AT models [6-11], the strike and the lightning channel are represented by thin wires. The AT models were exclusively applied to an analysis of lightning strike to CN tower in

Toronto [4-11], or similar towers [10]. The formulation can be posed in the time [6], [11] or in the frequency domain [7-10], respectively. When the formulation is given in the frequency domain a Numerical Electromagnetic Code (NEC) is used for current distribution calculation. Initially, the ground was assumed to be perfectly conducting [6], while recently the finite ground conductivity, and the grounding parts of the tower were included in the analysis [8].

In this paper a direct lightning strike to the WT is analyzed. Wind turbine is represented by a simple perfectly conducting wire structure, consisting of tower and three blades, while the lightning channel is represented by a lossy vertical wire attached to the wind turbine. The lightning return stroke current is energized by an ideal voltage source at the tip of the wind turbine blade. The current distribution along the wind turbine and lightning channel is assessed by solving the set of Pocklington integro-differential equation in the frequency domain. The solution is based on the Galerkin-Bubnov variant of Indirect boundary Element Method (GB-IBEM). Although WT is considered to be a tall structure, to the best of authors knowledge, a direct lightning strike to the WT has not been analyzed so far.

II. MODEL PROPERTIES Geometry of the WT struck by lightning and its associated

thin wire antenna model is presented in Fig 1. The WT is represented by a simple configuration of four perfectly conducting wires representing the tower and three blades. The lightning channel is represented by a lossy vertical wire antenna, neglecting the effect of corona. The resistance per unit channel length was chosen to be 0.07 /m (the same value as [11] [12]). The wire radius was assumed to be 10 cm. For the sake of simplicity, the propagation of the current wave along the channel is equal to the speed of light. Although, the real speed of current propagation is a factor of two or three smaller than in this model, it has no effect on the distribution of current along the WT. The lightning return-stroke current is injected using voltage source on the tip of the WT blade.

Figure 1. Wind turbine striuck by lightning and related wire representation

A. Set of Pocklington Integro-differential equations for arbitrarily shaped wires The currents along the lightning channel and WT are

governed by the set of coupled Pocklington integro-differential equations for the wires of arbitrary shape in the frequency domain. This set of Pocklington equations is derived from Maxwell equations and by satisfying certain continuity conditions for the tangential field components of the electric field at the electrode surface [14].

The derivation of these integral equations is performed in similar manner like in the case of buried wires explained in [15]. For the sake of completeness, the formulation is outlined in this paper, as well. Thus, the set of Pocklington equations is given by:

20 0

'

20

'1

'

2 20

0

( ') ' ( , ') '

( ') * ( , *) '( )

( ') ( , ') '

( )

11,2,... ; ;4

n

W

n

n

n n m nC

Nn n in m nexc

sm Cn

sn m nC

S m

gW

g

I s s s k g s s ds

R I s s s k g s s dsE s

I s s G s s ds

Z I sk k

m N C Rj k

=

+ + + + + =

+ +

= = =

G G

G G

JGG

2 20k+

(1)

where I(s) is the induced current along the wire, ( )excsE s is the excitation function, SZ represents internal impedance per unit length of the wire, g0(s,s`) denotes the free space Green function, while gi(s,s`) arises from the image theory. These functions are given by:

0 0

00

( , ')jk Reg s sR

= ; 0 1

1

( , *)jk R

ieg s s

R

= (2)

and R0 and R1 is the distance from the source point and its image to the observation point, respectively. Furthermore, k0 and kg are propagation constants of air and lossy ground, respectively:

2 20 0 0k = (3)

2 2 20 0 0

gg efec rgk j

= =

(4)

where rg and g is the relative permittivity and conductivity of the ground, respectively, and is operating frequency. The third term ( , ')sG s s

JG contains the Sommerfeld integrals and is

constructed from the vector components for horizontal and vertical dipoles [15] [16]

( ) ( )

( ) ( )( , ') '

'

H H Hs z

V Vz

G s s x s G G G z

z s G G z

= + + +

+ +

JG JG G GJG G (5)

The vector components are given in [16].

At a junction consisting of two or more segments the continuity properties of the electric field has to be satisfied [15], which is ensured by applying the Kirchhoff current law and continuity equation.

B. Numerical solution The set of Pocklington integro-differential equations (1) is

numerically handled by means of the Galerkin-Bubnov variant of Indirect Boundary Element Method (GB-IBEM).

The unknown current ( )enI along the n-th wire segment is expressed by the sum of a finite number of linearly independent basis functions fni, with unknown complex coefficients Ini:

( ) { } { }1

' ( ')n

Ten ni ni n n

iI s I f s = f I

=

= (6) using the isoparametric elements yields:

( ) { } { }1

( )n

Ten ni ni n n

iI I f = f I

=

= (7) where n is the number of local nodes on the element.

A linear approximation over a boundary element along n-the wire is used in this work and the corresponding shape functions are given by:

1 21 1

2 2f f += = (8)

as this choice was proved to be optimal in modeling various wire structures [14].

Applying the weighted residual approach featuring the Galekin-Bubnov procedure the set of Pocklington equations is transformed into a system of algebraic equations.

Performing a certain mathematical manipulations, the following matrix equation is obtained:

[ ] { }i1 1

= 0, m =1,2,..., ; j =1,2,...,w nN N e

n w mjin i

Z I N N= =

(9) where Nw is the total number of wires, Nm is number of elements on the m-th antenna and Nn is number of elements on the n-th antenna. [ ] jiZ is the mutual impedance matrix for the j-th observation boundary element on the m-th antenna and i-th source boundary element on the n-th antenna.

Implementing isoparametric elements yields the following expression for mutual impedance matrix:

[ ] { } { }

{ } { }

{ } { }

{ } { }

1 1

01 1

1 120 0

1 11 1

1 11

20

1

' ( , ' ) ''

' ' ( , ' ) ''

' ( , * ) ''

* ' ( , * ) ''

e T n mnm m nj iji

T n mm n nm m nj i

T n minm m nj i

T n mm n inm m nj i

ds dsZ D D g s s d dd d

ds dsk s s f f g s s d d

d d

ds dsR D D g s s d dd d

ds dsR k s s f f g s s d d

d d

=

+

+

{ } { }

1

11 1

1 1

' ( , ' ) ''

T n mm snm m nj i

ds dss f f G s s d dd d

+

JG

(10)

Matrices {f} and {f'} contain the shape functions while {D} and {D'} contain their directional derivatives.

The frequency properties of voltage source at the top of the WT are determined by the desired current waveform at the base of the lightning channel and by the input impedances of the lightning channel and the WT [11].

Once calculating the current frequency spectrum, the time domain current distribution along the WT and lightning channel is obtained using the Inverse Fourier Transform.

III. NUMERICAL RESULTS All the results presented in the paper are calculated for the

input current waveform presented in Fig. 2. representing the current waveform at the lightning channel base when it is not influenced by the strike object. The current and current derivative peaks are assumed to be 4.7 kA and 25 kA/s, respectively [11]. The simulations were carried out for the different ground conductivities, i.e. = (PEC ground), = 0.01 S/m, =0.001 S/m and =0.0001 S/m. The relative permittivity of the ground was assumed to be r=10.

It should be noted that, for the sake of simplicity, grounding wires were not considered, as it was shown that they do not significantly affect the current distribution [9] [13].

0 2 4 6 8 101

0

1

2

3

4

5

I (kA)

t ( s)

Figure 2. The source current waveform specified using Heidlers function

The current is analyzed in the specific points along the WT, namely strike blade tip, middle of strike blade, middle of side blade, WT base as indicated in Fig. 3.

Lightning channel

Strike blade tip

Middle of the strike blade

Middle of the side blade

WT base

Figure 3. Specific points for current analysis

Figure 4 shows current waveform for different points along the WT in the case of PEC ground. All the major reflections are clearly visible. Fig. 4. clearly shows differences between the current in different places on the WT, e.g. the maximum level of current in the side blades is more than five times smaller than in the strike blade. Also, transient in the side blades is much shorter than in other parts of WT.

0 1 2 3 4 5 6 7 8 9 102

0

2

4

6

8

10

12

I [kA]

t [s]

PEC ground

CurrentStrike blade tipMiddle of side bladeMiddle of strike bladeWT baseChannel 1km

Figure 4. Current waveforms on the different points of WT for PEC ground

Figures 5 8 show current waveforms for different ground properties on different points along the WT: strike blade tip, middle of strike blade, middle of side blade, WT base, respectively. Clearly, there differences between current waveforms for different ground conductivities are small which is consistent with the results published in [8] for the CN tower.

Figure 5 specifically shows all major reflection. Refl.1 represents first reflection from the wire junction (i.e. WT nacelle) and it occurs after 0.253 s. Refl. 2 represents

reflection from the ends of the side blades and it occurs after 0.507 s. A final reflection occurs from the ground after 0.78 s and it is tagged as Refl. 3. Due to secondary reflections current reaches maximum around 1.6 s. After that follows the plateau with small oscillations and stabilization of current.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

I [kA]

t [s]

Strike blade tip

PECr=10 =1e2 S/m

r=10 =1e3 S/m

r=10 =1e4 S/m

Refl. 1

Refl. 2

Refl. 3

Secondary refls.

Figure 5. Current on the strike blade tip for different ground conditions with major reflections

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9I [k

A]

t [s]

Middle of strike blade

PECr=10 =1e2 S/m

r=10 =1e3 S/m

r=10 =1e4 S/m

Figure 6. Current on the middle of strike blade for different ground conditions

0 1 2 3 4 5 6 7 8 9 102

1.5

1

0.5

0

0.5

1

1.5

2

I [kA]

t [s]

Middle of side blade

PECr=10 =1e2 S/m

r=10 =1e3 S/m

r=10 =1e4 S/m

Figure 7. Current on the middle of side blade for different ground conditions

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12I [k

A]

t [s]

WT base

PECr=10 =1e2 S/m

r=10 =1e3 S/m

r=10 =1e4 S/m

Figure 8. Current on the WT base for different ground conditions

IV. CONCLUDING REMARKS The paper presents an antenna theory model of wind

turbine stroke by lightning. The WT is represented by a simple wire structure, while the channel is modelled as a lossy vertical antenna energized by a voltage source placed on the tip of the blade. The current distribution along the structure is governed by the set of the coupled Pocklington integro-differential equations. The corresponding set of integral equations in the frequency domain is solved using GB-IBEM. Finally, transient response is computed via Inverse Fourier Transform (IFT). The numerical experiments were carried out for the ground with different conductivities.

The presented results show that finite ground conductivity has a minor effect on the current distribution along the struck WT, which is consistent with previously published papers dealing with CN tower. Furthermore, although the WT model is very simple, the waveforms of the calculated current include all the major reflections being physically plausible.

Future work will include much more realistic model including overall structure of the WT together with the grounding structure and also with lightning channel that has reduced propagation speed.

REFERENCES [1] F. Rachidi, M. Rubinstein, J. Montanya, J.H. Bermudez, Rodriguez R.

Sola, G. Sola, N. Korovkin: A Review of Current Issues in Lightning Protection of New-Generation Wind Turbine Blades, IEEE Transactions on Industrial Electronics, vol. 55, n. 6, 2008, pp. 2489-2496.

[2] V. A. Rakov, "Transient response of a tall object to lightning," IEEE Transactions on Electromagnetic Compatibility, vol. 43, pp. 654-61, 2001.

[3] V. A. Rakov and M. A. Uman, "Review and evaluation of lightning return stroke models including some aspects of their application," IEEE Transactions on Electromagnetic Compatibility, vol. 40, pp. 403-26, 1998.

[4] F. Rachidi, V. A. Rakov, C. A. Nucci, and J. L. Bermudez, "The Effect of Vertically-Extended Strike Object on the Distribution of Current Along the Lightning Channel," Journal of Geophysical Research, vol. 107, pp. 4699, 2002.

[5] D. Pavanello, F. Rachidi, V. A. Rakov, C. A. Nucci, J. L. Bermudez, Return stroke current profiles and electromagnetic fields associated with lightning strikes to tall towers : Comparison of engineering models. Journal of Electrostatics, 65, 316-321. 2007

[6] A. S. Podgorski and J. A. Landt, "Three dimensional time domain modelling of lightning," IEEE Transactions on Power Delivery, vol. PWRD-2, pp. 931-938, 1987.

[7] E. Petrache, F. Rachidi, D. Pavanello, W. Janischewskyj, A. M. Hussein, M. Rubinstein, V. Shostak, W. A. Chisholm, and J. S. Chang, "Lightning Strikes to Elevated Structures: Influence of Grounding Conditions on Currents and Electromagnetic Fields," presented at IEEE International Symposium on Electromagnetic Compatibility, Chicago, 2005.

[8] E. Petrache, F. Rachidi, D. Pavanello, W. Janischewskyj, M. Rubinstein, W. A. Chisholm, A. M. Hussein, V. Shostak, and J. S. Chang, "Influence of the finite ground conductivity on the transient response to lightning of a tower and its grounding," presented at 28th General Assembly of International Union of Radio Science (URSI), New Delhi, India., 2005.

[9] A. S. Podgorski and J. A. Landt, "Numerical analysis of the lightning-CN tower interaction," presented at 6th Symposium and Technical Exhibition on Electromagnetic Compatibility, Zurich, Switzerland, 1985.

[10] Y. Baba and M. Ishii, "Numerical electromagnetic field analysis of lightning current in tall structures," IEEE Transactions on Power Delivery, vol. 16, pp. 324-8, 2001.

[11] B. Kordi, R. Moini, W. Janischewskyj, A. Hussein, V. Shostak, and V. A. Rakov, "Application of the antenna theory model to a tall tower struck by lightning," Journal of Geophysical Research, vol. 108, 2003.

[12] R. Moini, B. Kordi, G. Z. Rafi, V.A. Rakov, A new lightning return stroke model based on antenna theory model, Journal of Geophysical Research, vol. 105, no. D24, pp. 29 69329 702, Dec. 2000.

[13] F. Rachidi, Modeling lightning return strokes to tall structures : recent development, VIII International Symposium on Lightning Protection. Sao Paulo, Brazil. 2005.

[14] D. Poljak, Advanced Modeling in Computational Electromagnetic Compatibility, John Wiley & Sons, New York, 2007.

[15] D. Cavka, B. Harrat, D. Poljak, B. Nekhoul, K. Kerroum, K. EK. Drissi. Wire Antenna Versus Modified Transmission Line Approach To The Transient Analysis Of Grounding Grid, Engineering Analysis with Boundary Elements, Vol. 3, 2011, pp 1101-1108.

[16] G. J. Burke, E. K. Miller, Modeling Antennas Near to and Penetrating a Lossy Interface, IEEE Trans. on Antenna and Propagation, Vol. AP-32, No. 10, pp 1040-1049, October 1984.

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